If are non–zero, non-coplanar vectors, prove that the following vectors are coplanar :

and

Vectors parallel to the same plane, or lie on the same plane are called coplanar vectors

The three vectors are coplanar if one of them is expressible as a linear combination of the other two.

We have been given that, , and .

We can form a relation using these three vectors. Say,

Compare the vectors , and . We get

5 = 7x + 3y …(1)

6 = –8x + 20y …(2)

7 = 9x + 5y …(3)

Solving equations (1) and (2) for x and y.

Equation (1), 7x + 3y = 5

Equation (2), –8x + 20y = 6

Multiply equation (1) by 8 and equation (2) by 7, we get

7x + 3y = 5 [× 8

–8x + 20y = 6 [× 7

We get

⇒ 164y = 82

Put in equation (2), we get

⇒ –8x + 10 = 6

⇒ –8x = 6 – 10

⇒ –8x = –4

⇒ 8x = 4

Substituting and in equation (3), we get

7 = 9x + 5y

Or 9x + 5y = 7

⇒ 14 = 7 × 2

⇒ 14 = 14

∵, L.H.S = R.H.S

⇒ The value of x and y satisfy equation (3).

Thus, , and are coplanar.